Study of the crossing number associated with strong product of path with cycle and triangular snake graph

In 2018, Ouyang et al. presented the first efforts related to the crossing number of strong product of the path Pm to the cycle Cn. They proved that cr(P2 ⊠ Cn) = n for n ≥ 3 together with introducing a general conjecture as follows: cr(Pm ⊠ Cn) = (m - 1)n: ∀ m, n ≥ 3. Here, we prove that Ouyang et al. conjecture is also true for n = 3 and m ≥ 3, together with exhibiting an optimal drawing of it. Furthermore, we start to study new case in relation to the strong product of path with triangular snake graph TSn by proving that cr(P2 ⊠ TSn) = 3⌊n/2⌋ for n ≥ 3.